Question

In the electronic spectrum of a molecule the intensity of the absorption line is proportional to the squared of two integrals Transition Intensity ~ [intergal ψ’*el(r)×ψ”el(r) . r dr]2 [intergal ψ’*vib(R)×ψ”vib(R) dR]2 a) What is r and what is R ? b) What are ψ’*el(r) and ψ”el ? c) What are ψ’*vib(R) and ψ”vib(R) ? d) In the first integral the product of the wave functions ψ’*el(r)×ψ”el(r) is followed by r . Explain why the factor r is included in the integration. Where does it come from? e) How can you best describe the physical meaning of the second integral? f) Taking into account that Transition Intensity represents Transition Probability, explain why these two integrals are multiplied by each other (but not for example are added to each other, subtracted from each other, divided by each other etc.). What principle, assumption or approximation is behind the product (multiplication) of the two integrals?

Answer 1

a) Here r is the distance of the electron from the center of the atom. R is the separation between two atoms of the molecule.

b)  ψ’*el(r) and ψ”el(r) is the initial and final electronic space wavefunction.

c) ψ’*vib(R) and ψ”vib(R) is the initial and final vibrational wave function.

d) the factor of r comes from the fact that the intensity of absorption line is dependant on molecular electric dipole moment μ = - er and the intensity of transition is given by the integral of the total wave function of the initial and final state multiplied by molecular electric dipole moment μ.